Towards studying the structural differences between the pion and its radial excitation
Xiang Gao, Nikhil Karthik, Swagato Mukherjee, Peter Petreczky, Sergey Syritsyn, Yong Zhao
TTowards studying the structural differences between the pion and its radial excitation
Xiang Gao,
1, 2, ∗ Nikhil Karthik,
3, 4, † Swagato Mukherjee, Peter Petreczky, Sergey Syritsyn,
5, 6 and Yong Zhao Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA Physics Department, Tsinghua University, Beijing 100084, China Department of Physics, College of William & Mary, Williamsburg, VA 23185, USA Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA RIKEN-BNL Research Center, Brookhaven National Lab, Upton, NY, 11973, USA (Dated: February 3, 2021)We present an exploratory lattice QCD investigation of the differences between the valence quarkstructure of pion and its radial excitation π (1300) in a fixed finite volume using the leading-twistfactorization approach. We present evidences that the first pion excitation in our lattice computationis a single particle state that is likely to be the finite volume realization of π (1300). An analysiswith reasonable priors result in better estimates of the excited state PDF and the moments, whereinwe find evidence that the radial excitation of pion correlates with an almost two-fold increase inthe momentum fraction of valence quarks. This proof-of-principle work establishes the viabilityof future lattice computations incorporating larger operator basis that can resolve the structuralchanges accompanying hadronic excitation. I. INTRODUCTION
The parton structure of pion has garnered both exper-imental [1–10] as well as theoretical efforts [11–20]. Abetter determination of the quark structure of pion isalso part of the goals of upcoming experimental facili-ties [21, 22]. In addition to experimental determinations,due to the recent breakthroughs in computing partonstructure using the Euclidean lattice QCD simulationsvia leading-twist perturbative factorization approaches(Large Momentum Effective Theory [23, 24], short-distance factorization of the pseudo distribution [25, 26],current-current correlators [27–29], which has also beendubbed as good lattice cross sections [29], and Refs [30–34] for extensive reviews on the methodology), the va-lence quark structure of pion has also been investigatedfrom first-principle QCD computations [35–42]. Thelarge- x behavior of the valence pion PDF has been anunresolved issue that has been approached using all theabove lines of attack, with the promise of being settledin the near future by lattice computations with finer lat-tices, realistic physical pion masses and with the usage ofhighly boosted pion states to reduce higher-twist effectsthat might be amplified [43–45] near x = 1.The considerable interest in the quark structure of pionis due to its special role as the Nambu-Goldstone bosonof chiral symmetry breaking in QCD. The grand goalof this research direction is to understand the aspectsof mass-gap generation in QCD via the quark-gluon in-teraction within the pion. The large- x behavior of pionPDF has been proposed to hold the key to make this con-nection (c.f., [46]). While the enigmatic aspect of QCDis the presence of nonvanishing mass-gap between the ∗ [email protected] † [email protected] vacuum and the ground-states of various quantum num-bers even in the chiral limit (except the pseudo-scalar,which is an exception), it is equally enigmatic that thereare non-zero mass-gaps amongst the excited states in thetower of excited spectrum as well. To contrast, if thetrace-anomaly was absent in QCD, there would not bemass-gaps between the vacuum and the various ground-states, nor between the excited states. Given the starkdissimilarity between the vanishing mass of a pion in thechiral limit and the nonvanishing masses of its excitedstates in the same limit, it is reasonable to expect anydifferences between the quark and gluon structures of theground-state pion and its excited states could help us un-derstand the mechanism behind spontaneous symmetry-breaking and the mass-gap generation better. In thisrespect, there have been prior lattice computations tostudy the decay constants of the pion and its radial ex-citation [47, 48], where the decay constant of the radialexcitation is expected to vanish in the chiral limit un-like that of the ground-state pion [49]. Closely relatedto the decay constant, the distribution amplitudes of thepion and its radial excitation have also been previouslystudied using the Dyson-Schwinger Equation [50]. Withthe lattice computation of PDFs now possible, a noveltheoretical research direction to study not just the differ-ences between the long-distance behaviors of the groundand the excited states, but to study the differences intheir internal structural properties is promising. In thisrespect, we should also point to a previous study [51]of the ∆ + baryon on the lattice, which differs by bothmass and angular momentum from that of the proton.Since there is also experimental thrust to understand ex-otic gluon excitations of mesons in Jefferson Lab 12 GeVprogram [52], studies as the present one on the partonstructure of simpler radial excitations, might be helpfulphenomenologically by providing a case to contrast theexotic transitions with.It is the aim of this paper to point to the possibility of a r X i v : . [ h e p - l a t ] F e b t min / a E ( P z ) a SS, P z = 0 E , N state = 2 t min / a E ( P z ) a SS, P z = 0.48 GeV E , N state = 2 t min / a E ( P z ) a SS, P z = 0.97 GeV E , N state = 2 t min / a E ( P z ) a SS, P z = 1.45 GeV E , N state = 2 t min / a E ( P z ) a SS, P z = 1.94 GeV E , N state = 2 t min / a E ( P z ) a SS, P z = 2.42 GeV E , N state = 2 FIG. 1. The dependence of the first-excited state energy E ( P ) on the range [ t min ,
32] used in the two-state fits to the SScorrelator is shown. The different panels are for six different values of momentum P = (0 , , P z ). For large t min , the best fitvalues have a tendency to approach the dispersion values, E = (cid:112) P z + M with M = 1 . studying the structural differences between the pion andits radial excitation [53], π (1300). In this paper, we willprovide reasonable evidences to justify that the excitedstate that we observe on the lattice shows properties of asingle particle state with similar mass to that of π (1300),a broad resonance state with decay-width of 200 to 600MeV, which has been rendered stable in the fixed finitevolume of this lattice computation. Then, we will showinteresting features in the excited state bilocal quark bi-linear matrix elements and the extracted PDFs and itsmoments, all under the justified hypothesis that the firstexcited state on the lattice is indeed π (1300). II. DETAILS ON TWO-POINT FUNCTIONANALYSIS AND EVIDENCES FOR π (1300) ASTHE FIRST-EXCITED STATE
In Refs [35, 37], we previously studied the valence PDFof pion at two fine lattice spacings of 0.04 fm and 0.06fm. In this section, we discuss the numerical evidences inthese previous computations that the first excited state,occurring in the spectral decompositions of the two-pointand the three-point functions, is likely to be a single par-ticle state, and that it corresponds to the first pion radialexcitation, π (1300). We will do this by first showing thatthe excited state energy obtained by the two- and three-state fits to the pion two-point function is consistent witha single particle energy-momentum dispersion relation.Then, we will notice that the mass of this state obtainedfrom the P z = 0 correlator lies close to 1 . π (1300), and the discrepancy is only about200 MeV. A source of this discrepancy could simply bethe heavier than physical pion mass used in this work. Another source could be that the first excited state iscomputed in a fixed finite volume, and it can differ fromthe pole mass of the actual resonance in the infinite vol-ume limit. Below, we elaborate further.In this work, we solely concentrate on the a = 0 .
04 fmlattice spacing ensemble used in [35], which consists of L t × L lattices with L t = L = 64. We used Gaussiansmeared-source smeared-sink set-up (SS), as well as thesmeared-source point-sink set-up (SP) to determine thetwo-point functions of pion, C ( t s ) = (cid:10) π ( P , t s ) π † ( P , (cid:11) . (1)In the above construction, we used momentum (boosted)quark smearing [54] to improve the signal for the boostedhadrons. We have discussed the details of the parame-ters used in the source-sink construction, as well as ouranalysis methods for the two-point function in our pre-vious publication Ref [35]. It is worth pointing out thatwe were able to obtain a visible signal for the first ex-cited state in the a = 0 .
04 fm computation, that we willdescribe in the next section, because the smearing radiusof the quark sources was kept constant in lattice unitsinstead of in physical units; namely, for the a = 0 .
06 fmensemble with an optimal tuning, the radius of Gaussiansource was 0 .
312 fm, whereas on the a = 0 .
04 fm ensem-ble, our choice resulted in a radius of 0 .
208 fm which issmaller than the optimal one.We analyzed the spectral content of the two-point func-tion through fits to the two- and three-state Ansatz;namely C ( t s ) = N state − (cid:88) i =0 | A i | (cid:16) e − E i t s + e − E i ( L t − t s ) (cid:17) , (2) . . . . . . . E ( P )( G e V ) | P | (GeV) E E . . . E ( P )( G e V ) | P | (GeV) E E E FIG. 2. Observation of particle-like dispersion for ground-state and first excited state in pion correlator. (Left) The energies offirst two excited states as extracted from two-state and three-state fits to the pion two-point function are shown as a function ofthe magnitude of spatial momentum | P | . The red circles are for the ground state pion using two-state fit to the SS correlator.The blue symbols are for E ; the filled ones correspond to estimates from two-state fits and the open ones to the estimates fromthree-state fits with priors based on the estimates from the SP correlator. The values of E from the two types of fits agreewell with the particle-like dispersion (cid:112) | P | + E (0) (blue band). (Right) The second excited state E from the three-statefits are shown in addition to E and E . with N state = 2 and 3 respectively. The amplitudes A i and the energies E i are the fit parameters in this analy-sis. The reason for using two different choices of sourceand sink is two fold; first, the SP correlator has a largercontribution from the excited state and second, to checkfor the consistency between the energies extracted us-ing the two independent set of correlators. We checkedfor the robustness of the fit parameters by varying therange of source-sink separation, t s ∈ [ t min , t max ], used inthe fits and by making sure that the parameters haveplateaued. In [35], we studied only pions boosted alongthe z -direction. For this work, we also used pions boostedwith spatial momenta P = ( P x , P y , P z ) with non-zero P y and P x for the two-point function analysis, and obtainedtheir ground state pion energy E as a function of P using two-state fits to both SP and SS correlators. Wewere able to isolate the ground state energy well usinga fit range shorter than t s ∈ [0 .
56 fm , a ], whose valueswere consistent between both the SS and SP correlators.The resulting values of the ground-state E ( P ) followedthe continuum dispersion relation, E ( P ) = M π + | P | , (3)with M π = 0 . | P | = 2 .
42 GeV on our fine lat-tice. Having demonstrated that the actual lattice resultsfor the ground state satisfied our expectations about asingle particle pion state, we simply used the values of E ( P ) from Eq. (3) to fix the values of E in the spec- tral decomposition in Eq. (2) and determined the otherfree parameters; namely the amplitudes of the groundand first excited state, and the energy of the first excitedstate.We determined the first excited state energy E ( P )using (1) two-state fits to the SS and SP correlators withfixed value taken from the dispersion relation for E , and(2) by using three-state fits to the SS correlator with fixed E and imposing a prior on E with the central value andwidth of the prior set to the best fit value of E and itserror obtained from the two-state fit to the SP correlator.In Fig. 1, we show the dependence of E on the fit range[ t min , a ]. Each panel corresponds to the six differentmomenta P = (0 , , P z ), and for each momentum, wehave shown the t min dependence of E from the two-state fit. The best fit values of E plateau for t s ≥ a .First, we notice that the best fit value of E ( P z = 0) =1 . E ( P ) expectedfrom a single particle dispersion relation with a mass M = E ( P = 0) = 1 . E indeed approach the expected continuumvalues for non-zero momenta. This shows that the firstexcited state is likely to be a single particle eigenstate,and not a pseudo single particle state that effectively cap-tures a continuum of multiparticle states. In the case ofpion, such a possible multiparticle excited state is a threepion state with zero angular momentum and with theirtotal isospin being 1. For our ensemble, we estimate theinvariant mass of such a state to be 0.9 GeV, which ismuch smaller than the first excited state we are finding.One possibility is that the Gaussian source we are usingdoes not have an overlap with the three pion state due toits vastly different delocalized spatial distribution com-pared to a localized single particle state. To summarizeour evidence for observing π (1300), we plot the energy-momentum dispersion relation for the ground state andthe first excited state in the left panel of Fig. 2. For E , we have shown its estimates from the two-state fitswith prior on E , and from three-state fits with prioron E and E as described above — these are shown asthe blue filled and open squares in the figure, and theycan be seen to agree well with each other. We find thatboth E and E agree with their respective single particledispersion curves. From the three-state fits with priorson E and E , we were able to estimate the second ex-cited state E , which must capture the tower of excitedstates above π (1300). In the right panel of Fig. 2, wehave shown these estimates for E as the black trianglepoints, and shown it in comparison to E and E . Wewill use results on the spectrum from the three-state fitin the further analysis of three-point functions.Having demonstrated that the first-excited state ob-served in our computation is likely to be the first ra-dial excitation of pion, we will henceforth work underthe assumption that this is indeed the case, and ask forthe properties of this excited state given this justified as-sumption. In the rest of the paper, we will refer to thefirst excited state in our lattice computation by π (cid:48) , ratherthan calling it as π (1300). This is because the mass ofthe first excitation on our lattice is not 1300 MeV, andfor the sake of brevity. We will also simply label the firstexcited state mass M as M π (cid:48) . III. EXTRACTION OF EXCITED STATEBILOCAL QUARK BILINEAR MATRIXELEMENT
In order to determine the required bilocal quark bilin-ear matrix elements of the boosted π and π (cid:48) , we com-puted the three-point function C ( z, τ, t s ) = (cid:68) π S ( P , t s ) O ( z, τ ) π † S ( P , (cid:69) , (4)with both the source and sink smeared. The bilocal op-erator involving quark and antiquark separated spatiallyby distance z is O ( z, τ ) = (cid:88) x (cid:20) u ( x + L ) γ t W z ( x + L , x ) u ( x ) − d ( x + L ) γ t W z ( x + L , x ) d ( x ) (cid:21) , (5)where x = ( τ, x ) with τ being the time-slice where theoperator is inserted, and the quark-antiquark being dis-placed along the z -direction by L = (0 , , , z ). The bilo-cal operator is made gauge-invariant by using a straightWilson-line W z constructed out of 1-HYP smeared gaugelinks. Since we are interested only in the parton distri-bution function in this paper, we used P = (0 , , P z ) thatis along the direction of Wilson-line for the three-pointfunction computations. We used, P z = 2 πaL n z , (6)with n z = 0 , , , , ,
5. The spectral decomposition ofthe three-point function C ( t s , τ ; z, P z ) = (cid:88) i,j A ∗ i A j h ij ( z, P z ) e − E i ( t s − τ ) − E j τ , (7)contains information on all the matrix elements between i -th and j -th states with pion quantum number h ij ( z, P z ) = (cid:104) E i , P z | O ( z ) | E j , P z (cid:105) . (8)We obtained the values of the amplitudes | A i | and theenergies E i from the analyses of C . We fixed themto the central values from the three-state fits. One canextract the matrix elements h ij by fitting the t s and τ dependence of C data to the spectral decompositionabove, with h ij being the unknown fit parameters. Inpractice, for the cross-terms such as A ∗ A h , we simplytreated the real part of such whole factors together asthe fit parameters, whereas for the diagonal terms onlythe magnitudes | A i | enter, and therefore, we were ableto resolve the diagonal matrix elements h ii without anyphase ambiguity.We implemented this analysis by first forming the stan-dard ratio R ( t s , τ ; z, P z ) ≡ C ( t s , τ ; z, P z ) C ( t s ; P z ) , (9)so that the leading term in this ratio as t s → ∞ is theground state matrix element h . In [35], we presenteddetailed analysis of the ratio R using both the two-stateand three-state fits. In that work, we found that a simpletwo-state fit was enough to obtain h which was consis-tent with a more elaborate three-state fit as well as withthe summation method. On the one hand, a simple two-state fit is not justified here, as we are interested in thefirst excited state, and therefore, at least one more stateother than the first excited state should be included inthe analysis. On the other hand, a full three-state analy-sis involving 9 independent fit parameters will make thedeterminations of h noisy. Therefore, we experimentedwith variations of the three-state fit by reducing the num-ber of parameters in the fit and by imposing prior on the − . . . . . .
20 2 4 6 8 10 12 14 16 18 n z = 3 a = 0 .
04 fm h ( z , P z ) z/a type-1type-2type-3 − . . . . . . .
40 2 4 6 8 10 12 14 16 18 n z = 4 a = 0 .
04 fm h ( z , P z ) z/a type-1type-2type-3 − . − . − . . . . . . .
40 2 4 6 8 10 12 14 16 18 n z = 5 a = 0 .
04 fm h ( z , P z ) z/a type-1type-2type-3 FIG. 3. The bare matrix element, h ( z, P z ), for the first excited state of pion is shown using three different types of three-statefits. The panels from left to right correspond to n z = 3 , , ground state matrix element h from the two-state fit.We first performed the full three-state fit with 9 param-eters, which we call as the fit of type-1. Then, in a fitof type-2, we imposed a prior on h keeping all other fitparameters of the full state fit to be free; for the priorand its width we took the value of h and its statisticalerror from the two-state analysis of the three-point func-tion. In a fit of type-3, in addition to imposing the prioron h , we also assumed that we can ignore the second-excited state matrix element h , thereby reducing thefit parameters to 8 (or effectively 7, due to the prior).In all the three Ansatze, we kept all the matrix elementswhich involved the first excited state.In Fig. 3, we show π (cid:48) matrix element, h , as a func-tion of z at the three largest momenta. The differentcolored symbols are the extrapolations using the abovethree types of three-state Ansatz. For n z = 3, the type-1, nine-parameter three-state fit actually performs betterthan when constraints are imposed. However, this is nottrue at the higher n z = 4 , π (cid:48) mass.For n z = 3, the type-2 fit results in noisier estimates of h compared to type-1, whereas the type-3 fit results areconsistent with type-1 results with a slight reduction inerrors. Therefore, we find that the type-3 Ansatz leads toa reasonable reduction in the statistical errors with onlythe assumption that h matrix element can be ignored.In fact, from the unconstrained type-1 fits, we found thatthe resulting values for h were consistent with zero andit was merely making the results noisier. Therefore, theusage of type-3 Ansatz to obtain better estimates of h seems to be justified. We tried reducing the number ofparameters further by ignoring the cross-terms h and h , but it resulted in unreasonably ultra-precise estima-tions of h , showing that such constraints rule out mostof the parameter space — it would have been a positiveoutcome if there was a strong theoretical underpinningto ignoring the cross-terms, but in the absence of such ajustification, we avoided using such stricter constraints.From the fit results for n z = 5 shown in the rightmostpanel of Fig. 3, the usage of type-3 fit renders h at thismomentum usable. In the analysis of PDF that follows, we will use the values of h obtained using type-3 Ansatzfor the extrapolations, and we will also show results fromthe type-1 fits to contrast it against.The bilocal operator O needs to be multiplicativelyrenormalized [55–57]. The details pertaining to renor-malization as applied to our computations are describedin detail in [35, 37]. One possibility is to determinethe renormalization factors Z γ t γ t ( z, P R ) in the RI-MOMscheme [58–60] using off-shell quarks at momentum P R =( P Rz , P R ⊥ ), h Rπ (cid:48) π (cid:48) ( z, P z , P R ) = Z γ t γ t ( z, P R ) h ( z, P z ) Z γ t γ t (0 , P R ) h (0 , P z ) , (10)In addition to the multiplicatively renormalizing the op-erator, the ratio with the corresponding matrix elementat z = 0, helps reduce lattice corrections and any overallsystematical corrections, so that the expectation valueof the isospin charge of pion is 1 by construction at allmomenta. Another possibility is to form renormalizationgroup invariant ratios [26, 35, 61, 62] between the barematrix elements at two different momenta, M π (cid:48) π (cid:48) ( z, P z , P z ) = (cid:18) h ( z, P z ) h ( z, P z ) (cid:19) (cid:18) h (0 , P z ) h (0 , P z ) (cid:19) . (11)In the above ratio, the UV divergence of the operator isexactly canceled between the two bare matrix elements.Similar to an improved version of the RI-MOM schemewe defined in Eq. (10), the double ratio at non-zero z and z = 0 matrix elements in the above equation ensuresthat the isospin charge is normalized to 1. Since theUV divergence does not depend on the external states,the two matrix elements in the ratio need not be for thesame hadron. Therefore, we also construct the followingratio using the ground state pion matrix element as M π (cid:48) π ( z, P z , P z ) = (cid:18) h ( z, P z ) h ( z, P z ) (cid:19) (cid:18) h (0 , P z ) h (0 , P z ) (cid:19) . (12)For the above ratio, we take our determination of h ( z, P z ) from [35]. In the next section, we will discussthe relation of the above matrix elements to the PDF viathe one-loop leading-twist perturbative matching. . . . . . . . . . Z γ t γ t ( ) × h ( z = , P z ) P z (GeV) z = 0 M.E.Excited stateGround state FIG. 4. The RI-MOM renormalized local matrix element(mod Z q ≈
1) at z = 0 with ( P Rz , P R ⊥ ) = (1 . , .
34) GeVis shown as a function of P z for pion (blue) and π (cid:48) (red) asobtained from the three-state type-3 fit. The wrap-aroundeffect present in the three-point function at P z = 0 is notaccounted for in the plot. Before performing any double ratio, we can use the z = 0 renormalized matrix element to perform a simplecross-check. The pion source π ( P z , t s ) can excite only oneunit of the isospin charge, and hence each of the statesthat occurs in the spectral decomposition of the pion two-point function will carry unit isospin (up to terms due towrap-around effects, which are negligible for heavy ex-cited states). Therefore, measuring the isospin of ourfirst excited state before imposing any normalization con-dition serves a cross-check of the excited state extrapola-tions. In Fig. 4, we show Z γ t γ t ( z = 0 , P R ) h ( z = 0 , P z )as a function of P z after renormalization in RI-MOMscheme. It is the isospin charge modulo the quark wave-function renormalization which is nearly 1 at this latticespacing [37]. At P z = 0, our ground-state matrix ele-ment determination suffers from 2% lattice periodicityeffects [35], that in turn affects all the fitted parametersin the three-state fit, particularly resulting in a value of Z γ t γ t h slightly larger than 1. At all other non-zero P z ,the extracted isospin of the first excited state is consis-tent with 1, lending more confidence in the reliability ofour extrapolations. IV. COMPARISON OF THE PDFS OF π AND π (cid:48) We used our estimates of h from the type-1 and type-3 fits to obtain the PDF of π (cid:48) . For this, we used thetwist-2 OPE expressions [61] corresponding to the renor-malized matrix elements described above. For the RI-MOM matrix element at renormalization scale P R , thetwist-2 expression is h Rπ (cid:48) π (cid:48) ( z, P z , P R ) = 1+ (cid:88) c RIn ( µ, P R , z ) (cid:104) x n (cid:105) π (cid:48) ( − iP z z ) n n ! , (13)where the sum above runs over only the even values of n for the valence PDF of the pion and its excitations due to the isospin symmetry. The 1-loop expression for theRI-MOM Wilson coefficients is given in [35] using resultsin [31, 63]. The terms (cid:104) x n (cid:105) π (cid:48) ( µ ) = (cid:82) x n f v ( x, µ ) dx arethe moments of the valence PDF f v ( x, µ ) of the firstexcited state in the MS scheme at factorization scale µ .We will consistently use µ = 3 . M π (cid:48) π (cid:48) ( z, P z , P z ) = 1 + (cid:80) c n ( µ z ) (cid:104) x n (cid:105) π (cid:48) ( − iP z z ) n n ! (cid:80) c n ( µ z ) (cid:104) x n (cid:105) π (cid:48) ( − iP z z ) n n ! , (14)using the expressions for the Wilson coefficients c n givenin [61, 64]. Here, we also present results using a variantof the ratio scheme M π (cid:48) π described in the last section,and it has the leading twist expression, M π (cid:48) π ( z, P z , P z ) = 1 + (cid:80) c n ( µ z ) (cid:104) x n (cid:105) π (cid:48) ( − iP z z ) n n ! (cid:80) c n ( µ z ) (cid:104) x n (cid:105) π ( − iP z z ) n n ! , (15)where the moments (cid:104) x n (cid:105) π are those of the groundstate pion. We take their values from our analysis ofpion on the same ensemble presented in [35]. Sincethe mass of π (cid:48) is about 1.5 GeV, we took care oftarget mass correction at leading twist by replacing( P z z ) n → ( P z z ) n (cid:80) n/ k =0 ( n − k )! k !( n − k )! (cid:16) M π (cid:48) P z (cid:17) k in the above ex-pressions [65, 66]. We work under the assumption thatany target mass correction that can occur at higher twistare negligible. In order to justify this further, we even-tually used only the matrix elements at the two highestmomenta corresponding to P z = 1 .
93 and 2.42 GeV aswe discuss below.We performed two kinds of analysis. In a model inde-pendent analysis, we fitted the renormalized matrix ele-ments spanning a range of P z > P z and z ∈ [ z min , z max ]using their respective leading twist expressions above,with the even moments (cid:104) x n (cid:105) π (cid:48) as the independent fit pa-rameters. In the second kind of model dependent analy-sis, we assumed a two-parameter functional form of va-lence excited state PDF, f v ( x ) = N x α (1 − x ) β , (16)and fitted the resulting moments (which are functions of α and β ) to best describe the z and zP z dependences ofthe data. This enabled us to reconstruct the x -dependentPDF. Since the data for the excited state is noisy, wecould not improve the above parametrization by addingadditional small- x terms, as we did for the pion in [35]. Inthe future, one needs to perform a similar analysis withmultiple functional forms of the PDF Ansatz to quantifythe amount of systematic error. The nomenclature followed in this paper is such that (cid:104) x n (cid:105) is the n -th moment. − . − . . . . . .
20 1 2 3 4 5 6 7Type-3 P z = 0 .
48 GeV M π , π ( z , P z , P z ) zP z P z = 1 .
45 GeV P z = 1 .
93 GeV P z = 2 .
41 GeV − . − . . . . . .
20 1 2 3 4 5 6 7Type-3 P z = 0 .
48 GeV M π , π ( z , P z , P z ) zP z P z = 1 .
45 GeV P z = 1 .
93 GeV P z = 2 .
41 GeV − . − . . . . . .
20 1 2 3 4 5 6 7Type-3( P Rz , P R ⊥ ) = (1 . , .
34) GeV h R ( z , P z , P R ) zP z P z = 1 .
45 GeV P z = 1 .
93 GeV P z = 2 .
41 GeV − . − . . . . . .
20 1 2 3 4 5 6 7Type-3 P z = 0 .
48 GeV M π , π ( z , P z , P z ) zP z P z = 1 .
93 GeV P z = 2 .
41 GeV − . − . . . . . .
20 1 2 3 4 5 6 7Type-3 P z = 0 .
48 GeV M π , π ( z , P z , P z ) zP z P z = 1 .
93 GeV P z = 2 .
41 GeV − . − . . . . . .
20 1 2 3 4 5 6 7Type-3( P Rz , P R ⊥ ) = (1 . , .
34) GeV h R ( z , P z , P R ) zP z P z = 1 .
93 GeV P z = 2 .
41 GeV
FIG. 5. The excited state matrix elements renormalized in three different schemes (ratios M π (cid:48) π (cid:48) , M π (cid:48) π and RI-MOM h R from left to right) are shown as a function of zP z . The data points of the same momenta are shown using same colored symbols.The top panels show largest three momenta and the bottom one includes only the largest two. The bands are the fits usingthe respective leading-twist OPEs to data assuming a two-parameter functional form of the PDF. We first describe our reconstruction of the PDF usingthe two-parameter functional form. In Fig. 5, we put to-gether the renormalized bilocal matrix elements at dif-ferent fixed momenta and show them as a function of zP z .The left, middle and the right panels are in the two ratioschemes, M π (cid:48) π (cid:48) and M π (cid:48) π , and in the RI-MOM schemewith ( P Rz , P R ⊥ ) = (1 . , .
34) GeV respectively. We haveused the first non-zero momentum, P z = 0 .
48 GeV asthe reference momentum to construct the ratios, whichis slightly above Λ
QCD and also contributes minimally tothe statistical noise. In the top panels, the data from thethree highest momenta are shown, whereas in the bot-tom panels, only the two highest momenta, P z = 1 . The quantity zP z has also been referred to as the Ioffe-time [67],and the bilocal matrix element is also referred to as the Ioffe-time Distribution [25]. In the lack of a short-distance limit orinfinite momentum limit, the matrix element is common andexactly the same for both LaMET as well as the short-distancefactorization used in pseudo-PDF approach. Therefore, we referto the renormalized matrix elements as simply bilocal matrixelements, without any ambiguity. quark-antiquark separation z ∈ [2 a, . ideal analysis, where one wouldwant to keep range of z even smaller than what is usedhere. We skipped z = a to avoid the ( P z a ) lattice cor-rection [35]. Overall, the fits can be seen to performwell regardless of the momenta included in the analysis.However, upon a close inspection of the analyses in thetop-panel, we find that the evolution of the data with P z at different fixed zP z has opposite trends between thedata and the fits; namely, the central values of the datahave a decreasing tendency from P z = 1 .
45 GeV to 2.41GeV (albeit well within errors), whereas the fitted bandshave the opposite behavior. This indicates the presenceof possible higher twist corrections when matrix elementsat momentum P z = 1 .
45 GeV, which is comparable to themass of the excited state, is included in the analysis. Onthe other hand, in the lower-panel, the data at the twohighest momenta are compatible with each other and thefitted bands are also seen to be describing the data well.Therefore, to be cautious, we will simply include the dataat the two highest momenta in the analysis henceforth.In Fig. 6, we show the x -dependent valence PDF ofthe excited state, f v ( x ), that is reconstructed based onthe two-parameter Ansatz fits in the real-space shown asbands in the bottom panels of Fig. 5. The right panel ofFig. 6 is based on fits to the matrix elements obtainedusing type-3 extrapolation, whereas the left one is usingthe type-1 extrapolation. We have compared the PDF . .
520 0 . . . . f v ( x ) x type-1 M π π M π π h Rπ π π (GS) 00 . .
520 0 . . . . f v ( x ) x type-3 M π π M π π h Rπ π π (GS) FIG. 6. The valence parton distribution function f v ( x, µ ) of π (cid:48) at µ = 3 . f v ( x, µ ) = N x α (1 − x ) β . The results for the valence PDF obtained usingthe bilocal matrix elements in three different renormalization schemes are shown using the different colored bands. The resultsusing type-1 and type-3 matrix elements are shown on the left and right panels respectively. The valence PDF of pion asdetermined using the same ensemble and analysis methods as used for π (cid:48) is shown as the dashed curve for comparison. determinations as obtained from the fits to the matrixelements in the three different renormalization schemes.For comparison, the central value of the ground statepion PDF from the same ensemble [35] is shown as thedashed green line. First, the usage of type-1 extrapolatedmatrix elements results in very noisy PDF that cannot beused to find any hints of structural differences; within thelarge errors, the excited pion PDF is consistent with theground state PDF. On the other hand, the usage of type-3extrapolated matrix element does result in better deter-mined PDFs. Therefore, let us focus on the right panel ofFig. 6. The consistency amongst the estimates from dif-ferent renormalization schemes, which differ also in theirmatching formulas, is reassuring. Using M π (cid:48) π (cid:48) , we foundthe PDF is parametrized by { α, β } = { . , . } . Itis very clear that the PDF of the radial excitation is dif-ferent from the ground state — the excited state PDF isconsistently above the pion PDF starting from an inter-mediate x ≈ . x . There is a tendency in theexcited state PDF to vanish at small- x , but it is not con-clusive given the errors and also due to possibly largehigher-twist effects contaminating the small- x regime.Thus, the overall trend seems to be that the valence PDFof the radial excitation is shifted towards larger- x com-pared to the ground state valence PDF. This points tosmaller momentum fraction being carried by gluons andsea quarks in the radially excited state compared to thepion.In Fig. 7, we compare the first four valence PDF mo-ments of the radial excitation obtained using the model-dependent and model-independent analyses. The resultsfor π (cid:48) from various fitting procedures are shown on theleft part of the plot, and the values for the pion, takenfrom our previous work on the same ensemble [35], are shown on the right part of the figure. For the model-independent fits, we used the first three even moments (cid:104) x (cid:105) , (cid:104) x (cid:105) and (cid:104) x (cid:105) themselves as the fit parameters. Wealso imposed the inequality conditions between the va-lence moments as discussed in [35]. Similar to the PDFAnsatz fits, we present the results of the fits over a z -range of [2 a, . (cid:104) x (cid:105) and (cid:104) x (cid:105) are shown for them. The resultsfor the moments as inferred from the two-parameter fits,using the relation (cid:104) x n (cid:105) = (cid:82) x n f v ( x ) dx , are also shownfor π (cid:48) in Fig. 7; the results from fits to type-1 and type-3 M π (cid:48) π (cid:48) are labeled as C and D, whereas the ones fromfits to type-3 M π (cid:48) π matrix element are labeled as E. Itis comforting that the PDF Ansatz fits result in valuesof the even moments that are consistent with those fromthe model-independent fits. This also gives us the confi-dence in the indirect determination of the odd momentsvia this procedure. To justify this indirect method, tak-ing the case of pion where the phenomenological valuesof the odd moments are known, in [35], we found thatsimilar analysis via fits to PDF Ansatze resulted in val-ues of the odd moments that agreed reasonably well withthe phenomenological values.It is at once striking that the moments of π (cid:48) are largerthan that of the pion, especially in the case of the low-est two-moments (cid:104) x (cid:105) and (cid:104) x (cid:105) . Quantitatively, by takingthe values of (cid:8) (cid:104) x (cid:105) , (cid:104) x (cid:105) (cid:9) from the method “D” in Fig. 7,we see that they are { . , . } for π (cid:48) , which is tobe compared with { . , . } for the pion.This is the reason we observed the valence PDF of π (cid:48) tobe above that of the pion at higher values of x . There- . . . . . π π A B C D E h x ih x ih x ih x i h x n i FIG. 7. The lowest four valence PDF moments of the pionradial excitation π (cid:48) is compared with those of the ground statepion π . For the radial excitation, both the even and oddmoments extracted from the two-parameter PDF Ansatz fitare shown along with direct model independent estimates ofeven moments: (A) model-independent fit to type-1 M π (cid:48) π (cid:48) .(B) model-independent fit to type-3 M π (cid:48) π (cid:48) . (C) PDF Ansatzfit to type-1 M π (cid:48) π (cid:48) . (D) PDF Ansatz fit to type-3 M π (cid:48) π (cid:48) .(E) PDF Ansatz fit to type-3 M π (cid:48) π . The dashed line connectthe central values of π (cid:48) moments to that of π , to aid the eye. fore, at a scale of 3 . π (cid:48) momen-tum fraction comes from gluons and sea quarks, whichforms a larger 56% component for the ground state pion.Thus, within the two-parameter PDF Ansatz analysis, itappears that the valence quarks carry almost twice themomentum fraction in the radial excitation of the pioncompared to its ground state. This link could simply bea correlation or perhaps be causal, which needs to beinvestigated using simpler models. V. CONCLUSIONS AND OUTLOOK
In this work, we presented a proof-of-principle compu-tation of the first excited state of the pion determinedin a fixed finite volume and at a fixed fine lattice spac-ing. We argued that the first excited state is most likelyto be a single particle state since its energy satisfies asingle particle dispersion relation. Also, the mass of thestate compares well with the central value of the exper-imentally observed radial excitation, which is however aresonance in the infinite volume limit. Given the ob-servations, we hypothesized that the first excitation onour lattice is that of the pion radial excitation, π (1300).With a reasonable reduction in the number of unknownparameters in the three-state fits to the three-point func-tion involving the bilocal quark bilinear operator, we wereable to extract the boosted π (1300) matrix elements. Weperformed a model-independent analysis to obtain theeven valence PDF moments, and used model-dependentPDF Ansatz fits to reconstruct the x -dependent valence PDF at a scale of µ = 3 . π (1300) consis-tently lies above that of pion for intermediate and large- x regions, thereby indirectly, implying a reduced role ofgluons and sea quarks in the excited state. (2) Quanti-tatively, the lower moments of π (1300) were about twicelarger than that of π .The present work was meant only as a pilot study to-wards understanding how the ground and excited statesof hadrons differ. Therefore, this study can be made morerigorous in at least three major ways — (1) One couldeither render the radial excitation to be stable single-particle state by using a larger unphysical pion mass(e.g., [51]), or one needs to perform a dedicated finite-size scaling study of the excited state PDF in order tomake connection with the actual resonance state in thethermodynamic limit. (2) Usage of larger operator ba-sis in two-point functions that will lead to a more so-phisticated spectroscopy of pion correlators leading to amore convincing determination of the first excited stateas well as its quantum numbers. (3) Incorporating simi-lar techniques for the three-point function for a reliabledetermination of the excited state matrix element with-out involving any reduction in number of fit parametersas done here.As a concluding remark, in the absence of a micro-scopic theory of the transition from pion to its radialexcitation, we propose the following momentum differen-tial as a useful quantity. To motivate the quantity, onecan consider a process, such as π (cid:48) → π + ( ππ ) S − wave , fora special instance with both π (cid:48) and π after the transi-tion are at rest in the lab frame, and the difference intheir masses carried by other product states. In such anartificially constructed experimental outcome, one couldask how the change, ∆ P + = ( M π (cid:48) − M π ) / √
2, com-pares to the change in the average momentum ∆ (cid:104) k + (cid:105) =( M π (cid:48) × (cid:104) x (cid:105) π (cid:48) − M π × (cid:104) x (cid:105) π ) / √ π (cid:48) and π . This motivates the construction of theLorentz invariant ratio, ζ = 2 M π (cid:48) (cid:104) x (cid:105) π (cid:48) − M π (cid:104) x (cid:105) π M π (cid:48) − M π , (17)as a measure to correlate the structural changes to thedifferences in the masses. Using ∆ M = 1 . ζ ranges from 0.78 to 0.99 giventhe variations within 1- σ errors on the first moments wediscussed above. Even if we discount a 2- σ variation,the fraction is at least 0.68. Even such a simple-mindedmodeling of the excitation tells us that the changes tothe dynamics of valence parton could play an major rolein exciting a pion. ACKNOWLEDGMENTS
We thank C. D. Roberts for helpful comments on thepaper. This material is based upon work supported0by: (i) The U.S. Department of Energy, Office of Sci-ence, Office of Nuclear Physics through the ContractNo. DE-SC0012704; (ii) The U.S. Department of En-ergy, Office of Science, Office of Nuclear Physics and Of-fice of Advanced Scientific Computing Research withinthe framework of Scientific Discovery through AdvanceComputing (ScIDAC) award Computing the Propertiesof Matter with Leadership Computing Resources;(iii)X.G. is partially supported by the NSFC Grant Number11890712. (iv) N.K. is supported by Jefferson ScienceAssociates, LLC under U.S. DOE Contract No. DE-AC05-06OR23177 and in part by U.S. DOE grant No.DE-FG02-04ER41302. 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